Another crucial property of polarization kernels is the scaling exponent.
Let us fix a binary discrete memoryless channel \(W\)  of capacity \(I(W)\)  and a desired block error probability \(P_e\) . Given \(W\)  and \(P_e\) ,
suppose we wish to communicate at rate \(I(W) - \Delta \)  using a family of \((n,k)\)  polar codes with kernel \(K\) . The value of \(n\) 
scales as \(O(\Delta ^{- \mu (K)})\) , where the constant \(\mu (K)\)  is known as the scaling exponent [1]}. The scaling exponent depends on the channel. Unfortunately, the algorithm of its computation is only known for the case of binary erasure channel [1]}, [3]}. It is possible to show [4]}, [5]} that there exist \(l\times l \)  kernels \(K_l\) , such that \(\lim _{l\rightarrow \infty }\mu (K_l)=2,\) 
i.e. the corresponding polar codes provide an optimal scaling behaviour.
Constructions of the kernels with good scaling exponent are provided in
[6]}, [7]}.
